ANOTHER PROOF OF THE HARDY INEQUALITY IN A
LIMITING CASE WITH THE QUASI-SCALE INVARIANCE
MEGUMI SANO
Abstract. We introduce a new scaling in the logarithmic Hardy inequality. Our inequality has the quasi-scale invariance under the scaling. By
using a scaling argument, we also show that there is no minimizer for an
associated minimizing problem.
1. Introduction
Let Ω be a smooth bounded domain with 0 ∈ Ω in RN (or Ω = RN ). The
classical Hardy inequality
(
)p ∫
∫
N−p
|u(x)| p
(1.1)
dx ≤
|∇u(x)| p dx
p
p
|x|
Ω
Ω
holds for all u ∈ W01,p (Ω), where N ≥ 2, 1 ≤ p < N. It is known that, for
1 < p, the best constant ( N−p
) p is not attained in W01,p (Ω). Furthermore, the
p
inequality (1.1) can be improved by adding remainder terms in (1.1) (see
[1], [3], [5], [6], [8], [9], [10], [11], [13], [16] and the references therein).
One of the novelties of the inequality (1.1) is its scale invariance under the
scaling
( x)
N−p
(1.2)
uλ (x) = λ− p u
λ
for λ > 0 when Ω = RN . Indeed, one can easily check that
∫
∫
∫
∫
|uλ (x)| p
|u(y)| p
p
p
|∇uλ (x)| dx =
dx
=
dy.
|∇u(y)| dy,
p
|x| p
RN
RN
RN |y|
RN
On a bounded domain case, the inequality (1.1) does not have its scale invariance under the scaling (1.2) due to change in the domain of integration
by the scaling. Indeed, one can see that
∫
∫
∫
∫
|uλ (x)| p
|u(y)| p
p
p
|∇uλ (x)| dx =
|∇u(y)| dy,
dx
=
dy.
p
|x| p
λΩ
Ω
λΩ
Ω |y|
Date: May 4, 2015.
2010 Mathematics Subject Classification. Primary 35A23; Secondary 26D15, 46E35.
Key words and phrases. Hardy inequality, limiting case, minimization problem.
1
2
MEGUMI SANO
However, if we admit change in the domain of integration, we can say that
the inequality (1.1) in a bounded domain also has scale invariance under the
scaling (1.2). Here we call this property as the quasi-scale invariance.
On the critical case p = N, the inequality (1.1) fails for every constant
and instead of (1.1) the inequality
(
)N ∫
∫ |u(x)|N
N−1
x N
∇u(x) · dx
(1.3)
dx ≤
N
|x|
|x|N (log eR )N
Ω
|x|
Ω
holds for all u ∈ W01,N (Ω), where R := sup x∈Ω |x| (see [1], [17], Proposition
7 in Appendix). We call (1.3) as the Hardy inequality in a limiting case.
It is also known that the best constant ( N−1
)N is not attained in W01,N (Ω) by
N
adding remainder terms in (1.3) (see [2], [15], Proposition 7 in Appendix).
On the other hand, the critical Hardy inequality
(
)N ∫
∫ N−1
x N
|u(x)|N
∇u(x) · dx
(1.4)
dx ≤
N
|x|
|x|N (log R )N
Ω
|x|
Ω
holds for all u ∈ W01,N (Ω) (see [12], [17]). It is also known that the best
constant ( N−1
)N is not attained in W01,N (Ω) (see [12], [16]).
N
In this paper, we focus on the Hardy inequality in a limiting case (1.3).
In author’s opinion, the attainability of ( N−1
)N in (1.3) have been showed
N
by using adding remainder terms in (1.3) or by using the attainability of
( N−1
)N in (1.4). The main aim of this paper is to provide another proof of
N
the attainability of ( N−1
)N in (1.3) by using the property which the inequality
N
(1.3) has. We do not need remainder terms of (1.3) and results of (1.4).
Note that the inequalities (1.3), (1.4) are not the invariant under the scaling uλ (x) = u(λx) due to the logarithmic term. However Cassani-Ruf-Tarsi
[4] and Ioku-Ishiwata [12] introduced the scaling
(
)
N−1
(1.5)
uλ (x) = λ− N u |x|λ−1 x
for λ > 0 when Ω is a unit ball. One can observe that the inequality (1.4)
has its scale invariance under the scaling (1.5) when Ω is a unit ball (see
[12]). Ioku-Ishiwata [12] proved that the best constant ( N−1
)N in (1.4) is not
N
attained in W01,N (Ω) when Ω is a ball. One of key tools in their proof is its
scale invariance in the inequality (1.4). However, our inequality (1.3) does
not have its scale invariance under the scaling (1.5). Instead of (1.5), we
introduce the following scaling
( )λ−1 

 |x|
N−1
x
(1.6)
uλ (x) = λ− N u 
eR
for λ > 0 to the Hardy inequality in a limiting case (1.3). One of key tools
in our proof is its quasi-scale invariance in (1.3) under the scaling (1.6).
THE HARDY INEQUALITY WITH THE QUASI-SCALE INVARIANCE
3
Before starting main results let us introduce the following notations: B(R)
is a ball centered 0 with radius R in RN , |A| denotes the measure of a set
1,N
(B(R)) is a class of
A ⊂ RN , ωN is the area of the unit sphere in RN , and W0,rad
1,N
radially symmetric functions which is contained in W0 (B(R)). Moreover,
we set
N
∫ ∇u(x) · x dx
|x|
Ω
(1.7)
C H :=
inf
.
∫
N
|u(x)|
0,u∈W01,N (Ω)
(
)N dx
eR
Ω N
|x| log
Note that C H =
as follows:
( N−1
)N
N
|x|
from Proposition 7 in Appendix. Our main result is
Theorem 1. The optimal constant C H is not attained for any bounded domain Ω.
Remark 2. The inequality (1.3) has the quasi-scale invariance under the
scaling (1.6) (see Proposition 8 in Appendix).
2. Proof of Theorem 1
For a while, let Ω = B(R). We assume that C H is attained by ũ ∈
W01,N (B(R)) and derive a contradiction. At first, we show the existence of
a radial minimizer of C H . This lemma can be proved by using the same
method as one of [12].
Lemma 3 ([12]). Let ũ ∈ W01,N (B(R)) be a minimizer of C H . Then there
1,N
exists a radial minimizer u ∈ W0,rad
(B(R)) of C H .
Proof of Lemma 3. Since ũ ∈ W01,N (B(R)) is a minimizer of C H , then the
forth inequality in (3.2) should be an equality:
∫
|ũ(x)|N−1
x · ∇ũ(x) dx
(
)N−1 |x|
B(R) |x|N−1 log eR
|x|
 N−1
) N1 ∫
(∫ N

 N
N
x N
|ũ(x)|

∇ũ(x) · dx 
dx .
=
eR N
N
N − 1 B(R) |x|
B(R) |x| (log |x| )
By the equality condition for the Hölder inequality, there exists a constant
α ∈ R such that
x
|ũ(x)|
(2.1)
|x| · ∇ũ(x) = α |x|(log eR )
|x|
holds for almost every x ∈ B(R). Since ũ is a minimizer of C H , we have
1
α = (C H ) N .
4
MEGUMI SANO
On the other hand, by using polar coordinates x = (r, θ) (r ∈ (0, R), θ ∈
S ) and Fubini’s theorem together with the fact ũ ∈ W01,N (B(R)), we obtain
∫ R
∫ R
|ũ(r, θ)|
N N−1
(2.2)
|∂r ũ(r, θ)| r dr < ∞,
dr < ∞.
eR N
0
0 r(log r )
N−1
for almost every θ ∈ SN−1 . Combining (2.1) and (2.2), we observe that there
exists θ0 ∈ SN−1 such that
∫ R
∫ R
|ũ(r, θ0 )|
N N−1
0<
|∂r ũ(r, θ0 )| r dr = C H
dr < ∞.
eR N
0
0 r(log r )
Here we put u(x) := ũ(|x|, θ0 ). Then we can observe that u satisfies
N
∫ ∫R
∫
x N N−1
∇u(x) · |x| dx
dr
N−1 dθ 0 |∂r ũ(r, θ0 )| r
B(R) CH = S ∫
= ∫
∫ R |ũ(r,θ )|N
|u(x)|N
0
(
) dx
dθ
N dr
N−1
eR
B(R) |x|N log eR N
S
0 r(log )
|x|
r
1,N
which implies that u ∈ W0,rad
(B(R)) is a radial minimizer of C H .
□
Next lemma is concerning the quasi-scale invariance of the inequality
(1.3) for radial functions. Recall that the scale invariance of the inequality
(1.4) plays a essential role for proving the nonexistence of minimizers of
(1.4) in [12]. In our inequality (1.3), the quasi-scale invariance of (1.3)
plays a essential role for the proof of Theorem 1.
1,N
Lemma 4. Let u = u(s) ∈ W0,rad
(B(R)) and uλ = uλ (r) be defined by (1.6).
1,N
1− λ1
Then uλ ∈ W0,rad (B(e R)) and there hold
∫
∫
1
e1− λ R
|∂r uλ (r)| r
N N−1
(2.3)
R
dr =
|∂ s u(s)|N sN−1 ds
0
0
and
∫
(2.4)
0
1
e1− λ R
|uλ (r)|N
dr =
r(log eRr )N
∫
0
R
|u(s)|N
ds.
s(log eRs )N
1,N
Especially, if u ∈ W0,rad
(B(R)) is a radial minimizer of C H , then uλ ∈
1
1,N
1,N
W0,rad (B(e1− λ R)) ⊂ W0,rad (B(R)) is also a radial minimizer of C H for all
0 < λ ≤ 1.
Proof of Lemma 4. Since (1.6), there holds
∫ e1− λ1 R
∫ e1− λ1 R
|∂r uλ (r)|N r N−1 dr = λ1−N
|∂r u(rλ (eR)1−λ )|N r N−1 dr.
0
0
THE HARDY INEQUALITY WITH THE QUASI-SCALE INVARIANCE
5
Now applying the change of variables s = s(r) = rλ (eR)1−λ (i.e. r = r(s) =
1
1
s λ (eR)1− λ ), we obtain
∫ e1− λ1 R
∫ R
N N−1
1−N
|∂r uλ (r)| r dr = λ
|u′ (s)|N (s′ (r) r(s))N−1 ds
0
0
∫ R
=
|u′ (s)|N sN−1 ds
0
which implies (2.3). On the other hand, by the change of variables s =
1
1
s(r) = rλ (eR)1−λ (i.e. r = r(s) = s λ (eR)1− λ ), we get
∫ e1− λ1 R
∫ e1− λ1 R
|uλ (r)|N
|u(rλ (eR)1−λ )|N
1−N
dr
=
λ
dr
r(log eRr )N
r(log eRr )N
0
0
∫ R
|u(s)|N
1−N
=λ
ds
1
eR N
0 λs( λ log s )
∫ R
|u(s)|N
=
ds
eR N
0 s(log s )
which yields (2.4).
□
Remark 5. In fact, we can prove Lemma 4 for even non-radially symmetric
functions (see Proposition
( ∂(x ,··· ,x 8) )in Appendix). However, in that case, we must
use the Jacobian J ∂(y11 ,··· ,yNN ) in (3.7) calculated by Ioku-Ishiwata [12]. In
present proof, we do not need the complicated Jacobian thanks to restrict
functions to radial ones.
1,N
Lemma 6. Let u ∈ W0,rad
(B(R)) be a nonnegative minimizer of C H . Then
1
u ∈ C (B(R) \ {0}) and u > 0 in B(R) \ {0}.
Proof of Lemma 6. For the sake of simplicity, we put
∫ R
∫ R
|v(r)|N
N N−1
E(v) :=
|∂r v(r)| r dr, G(u) :=
dr.
eR N
0
0 r(log r )
1,N
1,N
where v ∈ W0,rad
(B(R)). For v, ϕ ∈ W0,rad
(B(R)), we have
∫ R
d
′
E (v)ϕ := E(v + tϕ) =
|∂r v(r)|N−2 ∂r v(r)∂r ϕ(r)r N−1 dr
dt t=0
0
∫ R
d |v(r)|N−2 v(r)ϕ(r)
′
dr.
G (v)ϕ := G(v + tϕ) =
dt t=0
r(log eRr )N
0
By using this notations, the optimal constant C H is written by
E(v)
=
inf
CH =
inf
E(v).
1,N
1,N
v∈W0,rad
0,v∈W0,rad
(B(R)),G(v)=1
(B(R)) G(v)
6
MEGUMI SANO
By the method of Lagrange multiplier, a minimizer u satisfies
∫ R
∫ R
|u(r)|N−2 u(r)ϕ(r)
N−2
N−1
|∂r u(r)| ∂r u(r)∂r ϕ(r) r dr = C H
dr
r(log eRr )N
0
0
1,N
(B(R)). This means that u is a weak solution of
for all ϕ ∈ W0,rad

|u|N−2 u


in B(R)

−∆N u = C H (|x| log eR )N
|x|



 u=0
on ∂B(R)
where −∆N v :=div (|∇v|N−2 ∇v) is the N-Laplacian. Particularly, for ε > 0,
(
)−N
eR
(2.5)
the function |x| log
is bounded in B(R) \ B(ε).
|x|
Furthermore Sobolev embedding W 1,N (B(R)) ,→ Lq (B(R)) (1 ≤ ∀ q < ∞)
yields that
u ∈ Lq (B(R)) for all 1 ≤ q < ∞.
(2.6)
Thus we see that ∆N u ∈ Lq (B(R) \ B(ε)) for all 1 ≤ q < ∞ by (2.5) and
N2
(2.6). If we take q > N−1
, then u ∈ C 1 (B(R) \ B(ε)). (see [7]) Hence,
by applying the strong maximum principle for the distributional solution
u ∈ C 1 (B(R) \ B(ε)) to the inequality −∆N u ≥ 0 in B(R) \ B(ε), we obtain
u > 0 in B(R) \ B(ε). (see [14] Theorem 2.5.1.) Since ε > 0 is arbitrary, we
have proved u ∈ C 1 (B(R) \ {0}) and u > 0 in B(R) \ {0}.
□
At last, we shall prove Theorem 1 by using three Lemmas. Proof of Theorem 1 consists of two Steps. In Step 1, we show the nonexistence of minimizers of C H when Ω is a ball B(R). In Step 2, we show the nonexistence
of minimizers of C H when Ω is a general bounded domain.
Proof of Theorem 1. (Step 1) : First we will show the optimal constant
C H is not attained when Ω = B(R). Now we assume that ũ ∈ W01,N (B(R))
is the minimizer of C H and derive a contradiction. From Lemma 3, there
1,N
exists a radial minimizer u ∈ W0,rad
(B(R)) of C H . Furthermore we suppose
that u is nonnegative without loss of generality. Let RS > 0 be such that
1
RS < R. From Lemma 4, if we take λ > 0 as satisfying e1− λ R = RS , then
1,N
uλ ∈ W0,rad
(B(RS )) and there hold
∫ RS
∫ R
N N−1
|∂r uλ (r)| r dr =
(2.7)
|∂ s u(s)|N sN−1 ds
0
and
0
∫
(2.8)
0
RS
|uλ (r)|N
dr =
r(log eRr )N
∫
0
R
|u(s)|N
ds.
s(log eRs )N
THE HARDY INEQUALITY WITH THE QUASI-SCALE INVARIANCE
Here we put



uλ (r)
uλ (r) := 

0
7
if 0 < r ≤ RS
if RS < r < R.
1,N
(B(R)) is also a radial minimizer of C H . However, by using
Thus uλ ∈ W0,rad
Lemma 6, we have uλ > 0 in 0 < r < R. Since uλ (r) ≡ 0 in RS ≤ r ≤ R, this
is a contradiction. Therefore the optimal constant C H is not attained when
Ω is a ball.
(Step 2) : Next we will show the optimal constant C H is not attained when
Ω is a general bounded domain. We assume that C H is attained by v ∈
W01,N (Ω). Then we put



if x ∈ Ω
v(x)
v(x) := 

0
if x ∈ B(R) \ Ω.
Since v ∈ W01,N (B(R)), C H is attained by v ∈ W01,N (B(R)) when Ω is a ball
B(R). This contradicts to Step 1.
The proof of Theorem 1 is now complete.
□
3. Appendix
For the reader’s convenience, we give a proof of the inequality (1.3) with
its optimal constant.
Proposition 7. Let Ω ⊂ RN be a smooth bounded domain with 0 ∈ Ω and
N ≥ 2. Then the inequality (1.3):
(
)N ∫
∫ N−1
|u(x)|N
x N
∇u(x) · dx
dx ≤
eR N
N
N
|x|
Ω |x| (log |x| )
Ω
holds for all u ∈ W01,N (Ω), where R = sup x∈Ω |x|. Furthermore, the constant
( N−1
)N is optimal, namely there holds
N
N
∫ )N
∇u(x) · x dx (
|x|
N−1
Ω
(3.1)
CH =
inf
=
.
∫
|u(x)|N
N
0,u∈W01,N (Ω)
)N dx
(
eR
Ω N
|x| log
|x|
Proof of Proposition 7. First we will describe the simple proof of (1.3) by
F.Takahashi [17]. Since




N−1
x


=
for |x| , 0, R,
div 

)
(

N−1
 |x|N (log eR )N
 N
eR
|x| log |x|
|x|
8
MEGUMI SANO
we have


∫
∫


N
|u(x)|
1
x


N
dx =
div 
(
)N−1  |u(x)| dx
eR N
N

N−1 Ω
Ω |x| (log |x| )
|x|N log eR
|x|
(
)
∫
N−2
N
x
|u(x)| u(x)
=
− · ∇u(x) dx
)
(
N − 1 Ω |x|N−1 log eR N−1 |x|
|x|
∫
N−1
|u(x)|
N
x
· ∇u(x) dx
≤
(
)
N − 1 Ω |x|N−1 log eR N−1 |x|
|x|
 N−1
(∫ ) N1 ∫
N
 N

x N
N
|u(x)|

∇u(x) · dx 
(3.2)
dx .
≤
eR N
N
N −1 Ω
|x|
Ω |x| (log |x| )
Hence the inequality (1.3) holds for all u ∈ W01,N (Ω).
Next we will give a proof of (3.1) by using the quasi-scale invariance in
the Hardy inequality in a limiting case (1.3). Let 0 < ℓ ≪ 1 and N−1
<δ<
N
1. We define the function uℓ,δ as follows:
(
)

e δ


log
− 1,
0 ≤ r ≤ ℓR

ℓ
(
)δ
uℓ,δ (r) = 


 log eRr − 1,
ℓR ≤ r ≤ R.
where r = |x|, and



0, (
)
uℓ,δ (r) = 

−δ log eR δ−1 r−1 ,
r
′
0 ≤ r ≤ ℓR
ℓR ≤ r ≤ R.
1,N
Therefore, one can easily check that u ∈ W0,rad
(B(R)). Direct calculations
show that
∫ R
∫ N
′
∇u (x) · x dx = ω δN
|uℓ,δ (r)|N r N−1 dr
N
ℓ,δ
|x|
ℓR
B(R)
∫ R(
eR )Nδ−N dr
= ωN δN
log
r
r
ℓR
∫ 1(
)
e Nδ−N dr
= ωN δN
log
r
r
ℓ
∫ log eℓ
= ωN δN
t Nδ−N dt
1
((
)
δN
e )Nδ−N+1
(3.3)
= ωN
log
−1 .
Nδ − N + 1
ℓ
THE HARDY INEQUALITY WITH THE QUASI-SCALE INVARIANCE
On the other hand, we have
∫
|uℓ,δ (x)|N
)N dx
(
B(R) |x|N log eR
|x|
((
)N ∫ ℓR (
)
e δ
eR )−N dr
= ωN log
−1
log
ℓ
r
r
0
)
(
∫
N
(
R
∑
eR )(N−k)δ−N dr
N
k
log
+ ωN
(−1)
k
r
r
ℓR
k=0
(
)∫ 1(
∫ ℓ(
N
∑
e )−N dr
e )(N−k)δ−N dr
N
k
= ωN
log
+ ωN
(−1)
log
k
r
r
r
r
0
ℓ
k=0
(
)(
∫
N
∑
e )(N−k)δ ∞ −N
N
k
t dt
= ωN
(−1)
log
k
ℓ
log eℓ
k=0
(
) ∫ log e
N
∑
ℓ
N
k
(−1)
+ ωN
t(N−k)δ−N dt
k
1
k=0
(
)
N
∑
1 (
e )(N−k)δ+1−N
N
log
= ωN
(−1)k
k N−1
ℓ
k=0
(
)
)
((
N
∑
1
e )(N−k)δ−N+1
N
k
+ ωN
(−1)
−1
log
k (n − k)δ − N + 1
ℓ
k=0
(
(
Nδ
e )Nδ+1−N
= ωN
log
(N − 1)(Nδ − N + 1)
ℓ
(3.4)
)
N
N
(
∑
e )(N−k)δ+1−N ∑
+
Ck,N,ℓ log
+
Dk,N,δ
ℓ
k=1
k=0
where
(
Dk,N,δ
)
(N − k)δ
(N − 1)( (N − k)δ − N + 1 )
(
)
1
N
:= (−1)k
k (N − k)δ − N + 1
Ck,N,δ := (−1)
k
N
k
9
10
MEGUMI SANO
From (3.3) and (3.4), we obtain
N
∫ ∇uℓ,δ (x) · x dx
|x|
(N − 1)δN−1
B(R)
→
as ℓ → 0
∫
|uℓ,δ (x)|N
N
)N dx
(
eR
B(R) N
|x| log
|x|
(
N−1
→
N
(3.5)
)N
as δ →
N−1
.
N
Now applying Lemma 4, we have
N
N
∫
∫ x ∇uℓ,δ (x) · x dx
1
∇(uℓ,δ )λ (x) · |x| dx
|x|
B(R)
B(e1− λ R)
(3.6)
=
∫
∫
|uℓ,δ (x)|N
|(uℓ,δ )λ (x)|N
1
(
) dx
(
) dx
eR N
eR N
B(R) N
B(e1− λ R) N
|x| log
|x|
|x| log
|x|
1
for all λ > 0. Here, if we take λ > 0 as satisfying B(e1− λ R) ⊂ Ω, then
1,N
(uℓ,δ )λ ∈ W0,rad
(Ω). Consequently, from (3.5) and (3.6), we can observe
)N is optimal in the inequality (1.3) for any bounded
that the constant ( N−1
N
domain Ω.
□
Proposition 8. The inequality (1.3) has quasi-scale invariance under the
scaling (1.6).
( |x| )λ−1 (
( |y| ) λ1 −1 )
Proof of Proposition 8. Let y = eR
y and
x i.e.x = eR

( ) λ1 −1


|y|

Ωλ := 
x=
y


eR




y
∈
Ω
.



( ,··· ,x ) )
1
N
Ioku-Ishiwata [12] calculated the Jacobian J ∂(x
as follows:
∂(y1 ,··· ,yN )
 ∂x1 ∂x1
∂x1 
 ∂y1 ∂y2 · · · ∂y

N 
 ∂x2 ∂x2
(
)
( )( λ1 −1)N
∂x2 
 ∂y1 ∂y2 · · · ∂yN 
∂(x1 , · · · , xN )
1
 = |y|
(3.7) J
:= det  .
.
.
.
.

..
. . .. 
∂(y1 , · · · , yN )
eR
λ
 ..

 ∂xN ∂xN

N 
· · · ∂x
∂y1
∂y2
∂yN
Since
( )λ−1
|x|
x
x
1
N
=λ
∇y u(y) · ,
∇ x uλ (x) ·
|x|
eR
|x|
there holds
∫ ∫ ( )(λ−1)N N
x N
|x|
∇u (x) · x dx =
∇y u(y) · dx.
λ
λ
|x| eR
|x|
Ωλ
Ω
THE HARDY INEQUALITY WITH THE QUASI-SCALE INVARIANCE
11
Thus we obtain
)
∫ ∫ ( )(1− λ1 )N N
(
|y|
y N ∂(x1 , · · · , xN )
∇u (x) · x dx =
λ ∇y u(y) · J
dy
λ
|x| |y|
∂(y1 , · · · , yN )
Ω eR
Ωλ
∫ y N
∇u(y) · dy.
(3.8)
=
|y|
Ω
( |y| ) λ1 −1
On the other hand, by the change of variables x = eR
y, we have
(
)
∫
∫
|uλ (x)|N
|u(y)|N
∂(x1 , · · · , xN )
1−N
dy
(
)N dx = λ
(
)N J
N
1
∂(y1 , · · · , yN )
Ωλ |x|N log eR
Ω (eR)(1− λ )N |y| λ 1 log eR
|x|
λ
|y|
∫
N
|u(y)|
(3.9)
=
(
)N dy.
Ω |y|N log eR
|y|
Therefore we have proved Proposition 8 by (3.8) and (3.9).
□
4. Acknowledgment
The author would like to thank Professor Hidemitsu Wadade (Kanazawa
University) for organizing the RIMS workshop in March, 2015, which gave
her a chance to study this problem. She also would like to thank Professor
Futoshi Takahashi (Osaka City University) for his constant encouragement.
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12
MEGUMI SANO
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Department of Mathematics, Graduate School of Science, Osaka City University,
Sumiyoshi-ku, Osaka, 558-8585, Japan
E-mail address: m13saM0311@st.osaka-cu.ac.jp